Monday, April 1, 2019

Playful Math Ed Blog Carnival (aka Math Teachers at Play) #126

126 = 6*21 = 2*3*3*7


If you want to choose 4 chicks randomly from 9 total chicks, there are 126 ways to do it.


Students learn more if they make up the stories for story problems themselves. Can your students make up stories for these ways of making 126?

126 = 27 - 21 (difference of powers of 2)
126  = 42 + 52 + 62 + 72 (sum of consecutive squares) 
126 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 (sum of consecutive even numbers)


This blog carnival has evolved from being mainly contributions to being mainly items the blog host has discovered. Since my passion lately is geometry, this issue is dedicated to geometry. (Which of the 3 ways of making 126 above has a geometric interpretation? Hint: There's a picture of it here... somewhere...)



Constructions
I have been intrigued for the past few years with Archimedes' method of finding pi. He didn't have the square root symbol, so he approximated using fractions, getting pi between 3 10/71 and 3 1/7. But if we follow his steps, and keep the square roots, we get a lovely pattern for our answer.You can try it. Construct a hexagon in a circle. If the radius of the circle is 1, then the hexagon's perimeter is 6. Perimeter over diameter = 6/2 = 3. Now create a dodecagon (12-sided polygon) from the hexagon. You can find the side lengths from repeated use of the Pythagorean theorem, and then find perimeter over diameter. Your result will be closer than for the hexagon. You can repeat this process until a pattern emerges.


If you want to get better at geometric construction (straightedge and compass style), play with it at sciencevsmagic or euclidthegame.

You can improve your geometric reasoning skills with the puzzles in Geometry Snacks (and More Geometry Snacks), by Ed Southall and Vincent Pantaloni. There are more puzzles at his blog. If you like them, the book is a treasure trove.

Because I've fallen in love with geometry, I decided to teach it this summer, for the first time ever. So I'm doing a lot to prepare. Henri Picciotto is an expert geometry teacher who graciously offered me his time over breakfast. He advised me to download his Geometry Labs book (free) from his Math Ed Page site. There is so much more there than this. But this alone was a huge gift. I think it may transform my course.


I've been collecting geometry mysteries. Medians are the lines from midpoints of the sides of a triangle to the opposite vertices. The 3 medians seem to always cross at one point. Why is that? I tried for weeks to prove it, and just couldn't. I finally gave up and looked at the proof. (And told my students how much fun I had failing!)   I then found another proof that followed a very different path. Can you prove it?

Here's a simpler mystery: If you make a 5-pointed star (perfectly even, I can't do that without digital help...), what is the angle at each point?


One of my favorites for seeing the geometry in math topics you didn't know were geometric is  Magic Pi - math animations. I hate that they're only on Facebook because I am not comfortable linking to facebook in class. But they are amazing. (I linked to one that's pure geometry. So cool.) They apparently do most of their animations in geogebra. I am a complete novice next to them. Here's a geogebra sketch I made today. It might be my first in their 3D mode.





Making Your Own Math
At the beginning, I mentioned having students make up their own story problems.  Here's a lovely post from Arithmophobia No More about just that. Here's another angle on teaching story problems, from Jen at Math State of mind. Leaving out the numbers helps students to slow down.

This blog post, by Amy at When Life Gave Us Lemons, is about her son making up his own math games. And John Golden has a whole class make up variations on a game he shared with them.


Denise Gaskins, founder of this carnival, pulls together so many books and ideas I love in this post. I don't know how she does it! The (surface) topic is fractions, but more than that, it made me think about how we can help students learn by saying less. The video she includes, with a teacher asking the two boys questions, and never telling them they're wrong, is fabulous. One of the commenters at Denise's post linked to a discussion of his own with a student. And that made me think about Bob Kaplan's guide to 'becoming invisible' (or not giving away the math). (What math delights have you found lately by following your nose? Bunny hops rock!)



You can check out the Carnival of Mathematics here. And if you'd like to host this carnival (we need help next month!), you can learn more and sign up here.



 

Friday, March 22, 2019

Coming soon: Math Teachers at Play (aka Playful Math Ed) blog carnival

I'll be posting the blog carnival here sometime late next week. Right now I'm beginning to gather links to lovely, playful math ed posts (and sites and videos and ...). If you know of something I should include, please email me at mathanthologyeditor on gmail.



Tuesday, February 12, 2019

When Math Tells a Story




On the Living Math Forum group, I claimed that Algebra 1 tells a story more than Algebra 2 does. N asked me to explain what I had in mind. Here's my reply (with a few revisions):

Lately I've been saying this sort of thing in my pre-caclulus class, not about the course, but about individual equations. We say math is a language. If it is, then we should be able to tell stories in it. Each equation makes a statement, and sometimes those statements tell stories.

The equation for a circle is (x-h)2 + (y-k)2 = r2 . Many students see each equation like this as separate from any other equations/formulas they know. I try to get them to look at this deeply. From the structure of it (square plus square equals square), I see that it's really the Pythagorean Theorem. Why would something for right triangles show up in the equation of a circle?! (That blew me away a few years back. I've been teaching math for 30 years, but that question seemed deep.)

It's because our coordinate system has the two axes perpendicular to each other. So the distance from the x-coordinate of a point on the circle to the x-coordinate of the center is measured horizontally and the similar y distance is measure vertically. You can build a right triangle from the center to (almost) any point on the circle. The constant radius is the hypotenuse of that right triangle.

So this equation tells a little story.

How does a whole course tell a story?

Algebra is about solving equations and about graphing. We want to see how real life situations (anything with data that has two components, like time and height) can be represented with equations and with graphs. In Algebra 1 students learn to solve simple equations and to graph. And hopefully they learn how the two skills are connected. There's a bit more. Systems of equations allow us to model slightly more complex situations, using more variables. And in my (community college) Beginning Algebra course (which is pretty much equivalent to a high school Algebra 1), our grand finale (after factoring) is graphing and recognizing equations of parabolas.

Near the beginning of the course, I introduce my favorite problem. I bought a tree and planted it. It was one foot tall at first. It grows two feet a year. Let's make a data table for height versus time, and a graph, and an equation. What does the input variable (let's use t instead of x) mean? What does the output variable (let's use h instead of y) mean? What does the slope mean? What does the y-intercept (or h-intercept) mean? The graph and the equation both tell the story of the tree. Linear growth is modeled with lines, which have equations of the form y = mx + b (or, in our tree story, h = 2t + 1). You can come at that from so many angles.

The course can tell the tree's story, or any story that can be told through data, graphs, and/or equations. It tells the story of using math to help us think quantitatively about problems we care about. (And of course there are plenty of things we care about that cannot be quantified.)

Sunday, February 10, 2019

Links to Good Math Posts

I am closing tabs, so that maybe my browser will move faster. Here are all the links I couldn't stand losing:

Thursday, January 10, 2019

Geometry and Visual Thinking Puzzle Collection - Help Needed






Math lovers, I need your help. (Math-likers welcome too. Math-haters who are into torturing themselves are also quite welcome, if you know any.) If you love puzzles, that might be enough to suck you in. I'm hoping...


I am pulling together a collection of problems meant to entice students who aren't all that into math. (So, relatively easy puzzles.) I think we focus too much on algebra and not enough on the visual skills that come along with geometry. So each problem in this collection includes visualizing. Most of them involve geometry.

I'm calling them PPODs - Puzzle Problem Of the Day. I plan to post them in my classrooms and in the math department of the community college I work at.

I have found quite a few good problems in Geometry Snacks, by Ed Southall and Vincent Pantaloni. (I bought the e-version of the book yesterday.) I just now discovered Ed's blog (solvemymaths.com), and found more goodies. My other favorite source is the beautiful by-hand drawings done by Catriona Shearer, @cshearer41. I couldn't find who to credit for about a quarter of the problems I've collected.

I'm aiming for 60 problems, one for (almost) each day (M-Th) of the 16-week semester. I have about 50 already. What I need is help determining the difficulty level of each one. I need volunteers to help me figure that out. I'm using a scale of 1 to 5 myself, and anything I'd call a 4 or 5, I'm throwing out. (I've gotten stuck on a few problems. Those are not in the collection, of course.)

If you're interested, I'll send you my collection. And then you post your ratings here.
  Just let me know in the comments if you'd like to check these out. (Or email me at mathanthologyeditor@gmail.com.) Also, I'm interested in your ratings and thoughts even if you just end up doing a few of the puzzles.
 
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